A statistical relation between the X-ray spectral index and Eddington ratio of active galactic nuclei in deep surveys

We present an investigation into how well the properties of the accretion flow on to a supermassive black hole may be coupled to those of the overlying hot corona. To do so, we specifically measure the characteristic spectral index, Gamma, of a power-law energy distribution, over an energy range of 2-10 keV, for X-ray selected, broad-lined radio-quiet active galactic nuclei (AGN) up to z similar to 2 in Cosmic Evolution Survey (COSMOS) and Extended Chandra Deep Field South (E-CDF-S). We test the previously reported dependence between Gamma and black hole mass, full width at half-maximum (FWHM) and Eddington ratio using a sample of AGN covering a broad range in these parameters based on both the Mg ii and H alpha emission lines with the later afforded by recent near-infrared spectroscopic observations using Subaru/Fibre Multi Object Spectrograph. We calculate the Eddington ratios, lambda(Edd), for sources where a bolometric luminosity (L-Bol) has been presented in the literature, based on spectral energy distribution fitting, or, for sources where these data do not exist, we calculate L-Bol using a bolometric correction to the X-ray luminosity, derived from a relationship between the bolometric correction and L-X/L-3000. From a sample of 69 X-ray bright sources (> 250 counts), where Gamma can be measured with greatest precision, with an estimate of L-Bol, we find a statistically significant correlation between Gamma and lambda(Edd), which is highly significant with a chance probability of 6.59x 10(-8). A statistically significant correlation between Gamma and the FWHM of the optical lines is confirmed, but at lower significance than with lambda(Edd) indicating that lambda(Edd) is the key parameter driving conditions in the corona. Linear regression analysis reveals that Gamma = (0.32 +/- 0.05) log(10)lambda(Edd) + (2.27 +/- 0.06) and Gamma = (-0.69 +/- 0.11) log(10)(FWHM/km s(-1)) + (4.44 +/- 0.42). Our results on Gamma-lambda(Edd) are in very good agreement with previous results. While the Gamma-lambda(Edd) relationship means that X-ray spectroscopy may be used to estimate black hole accretion rate, considerable dispersion in the correlation does not make this viable for single sources, however could be valuable for large X-ray spectral samples, such as those to be produced by eROSITA.